An important problem in radar, in sonar and in some communications systems is to develop samples of baseband in-phase and quadrature components of a given band-limited RF waveform. For example, any real signal can be considered to be the sum of a pair of complex signals, each complex signal being in the form s(t)=s.sub.i (t)+js.sub.q (t), with one complex signal having only positive frequency components and the other having only negative frequency components. One use of in-phase and quadrature components of a signal is to reduce the bandwidth which includes both positive and negative complex sidebands of a signal. A single sideband of the signal, that is the complex function of either positive or negative frequency components, can be suppressed as the other is frequency shifted to a dc baseband. The single remaining sideband, being a complex function, is defined in terms of in-phase and quadrature components of the initial band-limited waveform.
FIG. 1 illustrates the usual means by which in-phase and quadrature samples are produced. The spectra of two waveforms in the circuit are illustrated in FIG. 2. According to the Nyquist theorem, if a waveform whose Fourier transform is X(f) is band-limited such that there are no frequency components above a frequency w or below the frequency -w, the waveform can be entirely reconstructed from samples taken from the original waveform at a rate of at least 2w. In order to reduce the sampling rate required, a single sideband of a high frequency band-limited signal is conventionally shifted to the DC level as shown in FIG. 2B as the other is suppressed so that the waveform becomes band-limited within the frequencies -w.sub.2 and w.sub.2. When sampled, the sampled waveform spectrum is repeated periodically at a frequency equal to the sampling frequency as shown in FIG. 2C. It can be seen from a comparison of FIGS. 2B and 2C that so long as the sampling rate is at least as great as the bandwidth of the signal, that is 2w.sub.2, the center segment of the spectrum remains without overlap in the sampled spectrum and can be reconstructed by means of low pass filtering.
As shown in FIG. 2A, the original signal includes two sidebands centered at positive frequency f.sub.h and negative frequency -f.sub.h. The circuit of FIG. 1 is used to shift the upper sideband centered at f.sub.h to the baseband and to filter the lower sideband. Because the intended output of the circuit has a spectrum which is non-symmetric about zero frequency, the output must be complex. In the circuit of FIG. 1, the input x(t) having a spectrum X(f) centered at f.sub.h is applied to separate multipliers 22 and 24. The multipliers are conceptually like a complex multiplier whose fixed input is e.sup.-j2.pi.f h.sup.t ; therefore, in the complex output s(t)=s.sub.i (t)+js.sub.q (t), what had been the upper sideband of x(t) becomes centered at dc. The negative frequencies of x(t) become double-frequency terms and are removed by the low pass filters 26, 28. Therefore, s(t) has components only for -B/2&lt;f&lt;B/2 and, according to the Nyquist theorem, it may be sampled by sampling circuits 30, 32 with at least B complex samples per second to obtain a fully recoverable signal. The sampled signals are converted to digital samples by analog to digital converters 34, 36. The resultant spectrum Y(f) is a periodic signal of a period equal to the sampling frequency. In the following discussion the ideal realization of FIG. 1 will be taken as defining y.sub.n =i.sub.n +jq.sub.n, the desired in-phase and quadrature components.
FIG. 1 shows how non-ideal circuits can lead to a mismatch between i.sub.n and q.sub.n. It is often difficult to match the gains, phases and frequency responses of the two analog circuit chains which produce these two components.